Let's solve the given expression step-by-step:
The given expression is:
[tex]\[ \frac{(3xy + 2x)}{-\frac{x}{3}} \][/tex]
1. Identify the numerator and the denominator:
- Numerator: [tex]\(3xy + 2x\)[/tex]
- Denominator: [tex]\(-\frac{x}{3}\)[/tex]
2. Simplify the denominator:
To simplify the denominator [tex]\(-\frac{x}{3}\)[/tex], recall that dividing by a fraction is the same as multiplying by its reciprocal:
[tex]\[
-\frac{x}{3} = x \cdot \left( -\frac{1}{3} \right) = -\frac{1}{3} x
\][/tex]
3. Rewrite the original expression:
Now, we can rewrite the expression using the simplified denominator:
[tex]\[
\frac{3xy + 2x}{-\frac{x}{3}}
\][/tex]
4. Multiply the numerator and denominator by the reciprocal of the denominator:
Dividing by [tex]\(-\frac{x}{3}\)[/tex] is the same as multiplying by its reciprocal, which is [tex]\(-3/x\)[/tex]. Thus:
[tex]\[
\frac{3xy + 2x}{-\frac{x}{3}} = (3xy + 2x) \cdot \left( -\frac{3}{x} \right)
\][/tex]
5. Distribute the multiplication:
Distribute [tex]\(-\frac{3}{x}\)[/tex] across the terms in the numerator:
[tex]\[
\left( 3xy + 2x \right) \cdot \left( -\frac{3}{x} \right) = 3xy \cdot \left( -\frac{3}{x} \right) + 2x \cdot \left( -\frac{3}{x} \right)
\][/tex]
6. Simplify each term:
- For the first term:
[tex]\[
3xy \cdot \left( -\frac{3}{x} \right) = 3 \cdot y \cdot x \cdot \left( -\frac{3}{x} \right) = 3y \cdot (-3) = -9y
\][/tex]
- For the second term:
[tex]\[
2x \cdot \left( -\frac{3}{x} \right) = 2 \cdot x \cdot \left( -\frac{3}{x} \right) = 2 \cdot (-3) = -6
\][/tex]
7. Combine the simplified terms:
Combine the results to get the final expression:
[tex]\[
-9y - 6
\][/tex]
However, looking carefully at the multiplicative structure, we should recognize the form provided earlier:
8. Rewriting for structural clarity:
Simplify combining back, the structure is more compactly seen as:
[tex]\[
-3 (3xy + 2x) \cdot (\frac{1}{x}) = -3 \cdot (3xy + 2x) / x = -3(3xy + 2x)/x
\][/tex]
Therefore, the final step-by-step solution to the expression [tex]\(\frac{(3xy + 2x)}{-\frac{x}{3}}\)[/tex] results in:
[tex]\[
-3 \cdot \frac{(3xy + 2x)}{x}
\][/tex]
